Anisotropic Diffusion Model

• Anisotropic diffusion model is a transport model where diffusivity varies by direction, represented by tensors that encode geometry, stress, and data imbalance.
• It is applied across various domains, including astrophysics, image processing, and graph learning, providing direction-aware simulations and improved morphological fidelity.
• Numerical strategies such as FEM, closest-point methods, graph spectral techniques, and lattice Boltzmann discretizations address challenges in stability, identifiability, and scale consistency.

Searching arXiv for relevant papers on anisotropic diffusion models across several domains. Searching arXiv for "anisotropic diffusion model". An anisotropic diffusion model is a transport model in which spreading is direction-dependent and is therefore represented by a tensor, by distinct coefficients along preferred directions, or by direction- or class-specific schedules rather than a single scalar diffusivity. In the arXiv literature, this formulation appears in continuum PDEs for charged particles around pulsars, reaction–diffusion–mechanics systems, temporal Fokker–Planck equations, graph propagation operators, surface-intrinsic image filtering, phase-field models of surface diffusion, lattice-Boltzmann and SPH discretizations, and modern diffusion-based generative or discriminative learning systems (Yan et al., 2022, Cherubini et al., 2017, Elhag et al., 2022, Kong et al., 2024, Liu et al., 23 Feb 2026). Across these settings, the common structural change is the replacement of isotropic diffusion by a directionally structured operator whose coefficients encode geometry, stress, turbulence, microstructure, graph directionality, or data imbalance.

1. Mathematical form and defining structure

The canonical continuum form replaces the isotropic Laplacian by a tensor-weighted divergence. In the TeV-halo formulation, the electron phase-space density $n(\mathbf r,t,E_e)$ satisfies

$$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$

where $D_{\parallel}$ and $D_{\perp}$ are the diffusion coefficients along and perpendicular to the large-scale magnetic field $\mathbf B_0$ (Yan et al., 2022). In stress-coupled active media, the same structural idea appears as a constitutive law for a diffusion tensor,

$$d_{ij}(\sigma)=D_0\Bigl(\delta_{ij}+D_1\,\sigma_{ij}+D_2\,(\sigma^2)_{ij}\Bigr),$$

so that an initially isotropic tensor $D_0\mathbf I$ becomes anisotropic through the stress field $\sigma_{ij}$ (Cherubini et al., 2017). In temporal Fokker–Planck form, anisotropy is expressed by a diagonal tensor $\mathbf D(t)=\mathrm{diag}(D_1(t),\dots,D_n(t))$, with direction-specific coefficients $D_i(t)$ and drift $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$0 (Jones, 2013).

The same principle extends beyond classical diffusion PDEs. On graphs, isotropic heat diffusion is written as

$$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$1

while Graph Anisotropic Diffusion interleaves such linear diffusion with local anisotropic filters derived from a learned vector field, thereby producing multi-hop, direction-aware kernels (Elhag et al., 2022). In diffusion probabilistic models, anisotropy is not spatial but schedule-based: ADPM replaces the scalar noise schedule $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$2 by a class-dependent schedule $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$3, and the variational schedule-optimization framework generalizes scalar schedules to a matrix-valued path $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$4 that allocates noise across subspaces (Kong et al., 2024, Liu et al., 23 Feb 2026).

A concise comparison of representative formulations is useful because the same term denotes related but non-identical mathematical objects.

Setting	Anisotropic object	Representative form
TeV halos	Field-aligned tensor	$$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$5
Active deformable media	Stress-dependent tensor	$$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$6
Anomalous transport	Direction-specific coefficients	$$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$7
Graph learning	Direction-aware graph filters	diffusion step plus $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$8
DDPM variants	Class/subspace-dependent noise	$$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$9 or $D_{\parallel}$0

Positive definiteness or ellipticity is a recurring requirement. In the stress-driven model, positivity of the principal conductivities is needed for ellipticity (Cherubini et al., 2017). In the transient tensor-reconstruction model, $D_{\parallel}$1 is assumed for all $D_{\parallel}$2 (Luchesi, 15 Sep 2025). In anisotropic reaction–diffusion for brain tumours, the tensor $D_{\parallel}$3 is symmetric positive definite with spectrum contained in $D_{\parallel}$4 (Alnashri et al., 2020).

2. Mechanisms that generate anisotropy

The mechanisms producing anisotropy vary sharply by domain. In the TeV-halo model, sub-Alfvénic MHD turbulence with $D_{\parallel}$5 yields

$D_{\parallel}$6

and

$D_{\parallel}$7

so that slow effective diffusion is identified with cross-field transport rather than with an ad hoc diffusion suppression zone (Yan et al., 2022). In active deformable media, anisotropy is induced by nonlinear coupling: mechanics alter local microstructure, and objectivity motivates a tensor-valued function of the Cauchy stress, causing initially isotropic and homogeneous diffusion tensors to become inhomogeneous and anisotropic (Cherubini et al., 2017).

In image analysis, anisotropy is commonly data-driven. On curved surfaces, the diffusion tensor is constructed from a surface-intrinsic structure tensor and then specialized to Weickert-type edge-enhancing or coherence-enhancing forms, so that diffusion is suppressed across edges and promoted along coherent tangent directions (Naden et al., 2014). In the integrodifferential anisotropic diffusion model, anisotropy is created by multiscale integration: $D_{\parallel}$8 with scale-adaptive weights $D_{\parallel}$9 and contrast parameters $D_{\perp}$0, yielding a compact three-parameter model (Alt et al., 2020).

In learning systems, anisotropy is tied to sample imbalance or latent geometry rather than to physical space. ADPM increases the noise speed for underrepresented classes through

$D_{\perp}$1

motivated by a class-weighted generalization-error bound (Kong et al., 2024). The matrix-schedule framework instead uses orthogonal projectors $D_{\perp}$2 and monotone scalar functions $D_{\perp}$3 to define

$D_{\perp}$4

thereby assigning different noise levels to different subspaces (Liu et al., 23 Feb 2026). This suggests that “anisotropy” in current machine-learning usage includes any nonuniform allocation of diffusion across structured directions, whether those directions are geometric, semantic, or frequency-based.

3. Morphology, scaling laws, and observable consequences

Directional diffusivity changes geometry as much as it changes rates. In the TeV-halo model, a steady-state halo around a middle-aged pulsar is cylindrically symmetric about $D_{\perp}$5, but its projection onto the sky depends strongly on the viewing angle $D_{\perp}$6. The characteristic extents satisfy

$D_{\perp}$7

For $D_{\perp}$8, the halo is nearly circular; for $D_{\perp}$9, it appears noticeably elongated (Yan et al., 2022). The absence of apparent asymmetric morphology in the three detected halos is therefore not neutral evidence: the same paper argues that smaller viewing angles make halos more detectable and that this selection effect may explain why all three detected halos are consistent with being spherical (Yan et al., 2022).

In anomalous diffusion, anisotropy changes asymptotic scaling. With $\mathbf B_0$0, the directional variances obey

$\mathbf B_0$1

and the $\mathbf B_0$2-dimensional uncertainty volume scales as

$\mathbf B_0$3

In two dimensions this gives $\mathbf B_0$4, and in three dimensions $\mathbf B_0$5 (Jones, 2013). The model therefore unifies subdiffusion, normal diffusion, and superdiffusion direction by direction.

In active excitable media, anisotropy manifests in wavefront geometry and drift. Under sustained uniaxial stretch, conduction velocity along the stretch axis differs from the transverse direction, producing elliptical target patterns instead of circular ones; spiral waves drift toward regions of higher or lower stress-induced conduction, and larger $\mathbf B_0$6 or $\mathbf B_0$7 amplify these effects (Cherubini et al., 2017). In biofilms, anisotropy between radial water channels and azimuthal EPS transport yields a Green’s-function solution in polar coordinates; central symmetry makes isotropic and anisotropic responses coincide when the transmitter is at the center, whereas off-center transmitters produce greater diffusion peaks under anisotropy (Paramalingam et al., 2024). In anisotropic tempered diffusion, front propagation acquires a direction-dependent finite speed $\mathbf B_0$8, and jump fronts satisfy a Rankine–Hugoniot condition with velocity determined by the recession function of the potential $\mathbf B_0$9 (Calvo et al., 2020).

4. Numerical formulations and discretization strategies

Because anisotropic operators involve tensor contractions, cross-derivatives, geometry-aware gradients, or state-dependent coefficients, numerics are often as important as the PDE itself. Several discretization paradigms recur in the literature.

Numerical framework	Core idea	Representative source
Mixed-primal FEM	Direct approximation of stress and displacement with operator splitting	(Cherubini et al., 2017)
Closest-point method	Solve a 3D embedding PDE and re-extend to the surface	(Naden et al., 2014)
Graph spectral / implicit diffusion	Closed-form graph heat operator plus anisotropic filters	(Elhag et al., 2022)
B-TriRT LBM	Recover tensor diffusion via first-moment relaxation block $$d_{ij}(\sigma)=D_0\Bigl(\delta_{ij}+D_1\,\sigma_{ij}+D_2\,(\sigma^2)_{ij}\Bigr),$$0	(Zhao et al., 2019)
SPH full-Hessian model	Reconstruct the full Hessian with anisotropic kernels	(Tang et al., 2024)
GDM / HMM	Unified convergence framework for anisotropic reaction–diffusion	(Alnashri et al., 2020)

In the stress-driven reaction–diffusion–mechanics system, the numerical method uses unstructured triangular elements on $$d_{ij}(\sigma)=D_0\Bigl(\delta_{ij}+D_1\,\sigma_{ij}+D_2\,(\sigma^2)_{ij}\Bigr),$$1, Raviart–Thomas elements for stress, stabilized Brezzi–Douglas–Marini elements for displacement, continuous $$d_{ij}(\sigma)=D_0\Bigl(\delta_{ij}+D_1\,\sigma_{ij}+D_2\,(\sigma^2)_{ij}\Bigr),$$2 elements for $$d_{ij}(\sigma)=D_0\Bigl(\delta_{ij}+D_1\,\sigma_{ij}+D_2\,(\sigma^2)_{ij}\Bigr),$$3, backward Euler with $$d_{ij}(\sigma)=D_0\Bigl(\delta_{ij}+D_1\,\sigma_{ij}+D_2\,(\sigma^2)_{ij}\Bigr),$$4, and an operator-splitting loop that alternates reaction–diffusion and nonlinear elasticity solves (Cherubini et al., 2017). On curved surfaces, the closest-point method solves an embedding PDE in a narrow 3D band, then enforces the closest-point extension at every step; the resulting algorithm alternates a PDE step and an extension step and accommodates smooth, open, and triangulated surfaces (Naden et al., 2014).

For graph data, GAD offers two differentiable diffusion schemes: an implicit Euler step

$$d_{ij}(\sigma)=D_0\Bigl(\delta_{ij}+D_1\,\sigma_{ij}+D_2\,(\sigma^2)_{ij}\Bigr),$$5

and a truncated spectral expansion

$$d_{ij}(\sigma)=D_0\Bigl(\delta_{ij}+D_1\,\sigma_{ij}+D_2\,(\sigma^2)_{ij}\Bigr),$$6

These are then combined with anisotropic aggregators $$d_{ij}(\sigma)=D_0\Bigl(\delta_{ij}+D_1\,\sigma_{ij}+D_2\,(\sigma^2)_{ij}\Bigr),$$7 and $$d_{ij}(\sigma)=D_0\Bigl(\delta_{ij}+D_1\,\sigma_{ij}+D_2\,(\sigma^2)_{ij}\Bigr),$$8 derived from the Fiedler vector (Elhag et al., 2022). In lattice Boltzmann form, the B-TriRT model recovers the macroscopic tensor

$$d_{ij}(\sigma)=D_0\Bigl(\delta_{ij}+D_1\,\sigma_{ij}+D_2\,(\sigma^2)_{ij}\Bigr),$$9

so the anisotropic diffusion tensor is controlled directly by the first-moment relaxation block $D_0\mathbf I$0 (Zhao et al., 2019). In SPH, a Cholesky-based coordinate transform and a full-Hessian second-derivative model are used to approximate $D_0\mathbf I$1, enabling anisotropic contaminant transport, porous membrane diffusion, and cardiac electromechanics without spurious oscillations (Tang et al., 2024).

Inverse and reconstruction problems require additional structure. The transient coefficient-reconstruction model discretizes space by Chebyshev–Gauss–Lobatto collocation, produces a semi-discrete ODE

$D_0\mathbf I$2

and estimates the principal diffusivities $D_0\mathbf I$3 and $D_0\mathbf I$4 by a Levenberg–Marquardt procedure regularized with a smoothing operator $D_0\mathbf I$5 (Luchesi, 15 Sep 2025). The random-walk formulation on rectangular and hexagonal lattices arrives at explicit transition probabilities from a finite-volume discretization; the rectangular lattice imposes constraints on the diffusion tensor, whereas the hexagonal lattice removes that limitation (Filippini et al., 17 Oct 2025).

5. Domain-specific realizations

The term “anisotropic diffusion model” does not denote a single canonical model but a family of structurally related models adapted to very different observables.

In high-energy astrophysics, anisotropic diffusion is used to explain the small effective diffusion coefficients inferred for Geminga, Monogem, and LHAASO J0621+3755 without introducing a diffusion-suppressing zone. The paper adopts $D_0\mathbf I$6 at $D_0\mathbf I$7 GeV and explores $D_0\mathbf I$8 and $D_0\mathbf I$9, corresponding to $\sigma_{ij}$0 and $\sigma_{ij}$1 (Yan et al., 2022). In radiative transfer, anisotropic diffusion theory is linked explicitly to microscopic scattering properties through the diffusion tensor and a revised boundary condition; with those ingredients, diffusion solutions are reported to be in excellent agreement with Monte Carlo simulations in both steady-state and time-domain settings (Alerstam, 2013).

In continuum mechanics and interfacial dynamics, anisotropy enters through stress or surface energy. The stress-driven excitable-medium model targets mechano-electrical feedback in the heart, while the doubly degenerate anisotropic Cahn–Hilliard and ACH-IC models address anisotropic surface diffusion, weak versus strong anisotropy, energy dissipation, and improved conservation. ACH-IC replaces conservation of $\sigma_{ij}$2 by conservation of a more general $\sigma_{ij}$3, derives an evolution law from Onsager’s variational principle, and obtains second-order volume conservation together with energy dissipation (Salvalaglio et al., 2020, Zhou et al., 24 Jul 2025).

In image processing and visual computing, anisotropy is closely associated with edge preservation and coherence enhancement. Surface-intrinsic EED and CED extend Weickert’s flat-domain constructions to curved surfaces (Naden et al., 2014). The integrodifferential anisotropic diffusion model reports average PSNR values of $\sigma_{ij}$4 dB at $\sigma_{ij}$5, $\sigma_{ij}$6 dB at $\sigma_{ij}$7, and $\sigma_{ij}$8 dB at $\sigma_{ij}$9, exceeding PM, EED, and IID in the quoted comparisons (Alt et al., 2020). Reinforced Diffusion reframes denoising as a sequence of learned diffusion actions and reports, on BSD68 Gaussian denoising, $\mathbf D(t)=\mathrm{diag}(D_1(t),\dots,D_n(t))$0 dB at $\mathbf D(t)=\mathrm{diag}(D_1(t),\dots,D_n(t))$1, $\mathbf D(t)=\mathrm{diag}(D_1(t),\dots,D_n(t))$2 dB at $\mathbf D(t)=\mathrm{diag}(D_1(t),\dots,D_n(t))$3, and $\mathbf D(t)=\mathrm{diag}(D_1(t),\dots,D_n(t))$4 dB at $\mathbf D(t)=\mathrm{diag}(D_1(t),\dots,D_n(t))$5, with additional results for salt-and-pepper and Poisson noise (Qin et al., 30 Dec 2025).

In machine learning beyond restoration, anisotropic diffusion is used for graph representation and long-tailed classification. GAD reports ZINC MAE values of $\mathbf D(t)=\mathrm{diag}(D_1(t),\dots,D_n(t))$6 for GAD-implicit and $\mathbf D(t)=\mathrm{diag}(D_1(t),\dots,D_n(t))$7 for GAD-spectral without edge features, improving on the listed DGN baseline $\mathbf D(t)=\mathrm{diag}(D_1(t),\dots,D_n(t))$8; on QM9, GAD-spectral improves every reported property relative to DGN (Elhag et al., 2022). ADPM reports F1-score improvements of $\mathbf D(t)=\mathrm{diag}(D_1(t),\dots,D_n(t))$9 on PAD-UFES and $D_i(t)$0 on HAM10000 relative to the original diffusion probabilistic model, while maintaining head-class accuracy (Kong et al., 2024). The variational matrix-schedule framework reports FID improvements over baseline EDM across CIFAR-10, AFHQv2, FFHQ, and ImageNet-64 in all NFE regimes considered (Liu et al., 23 Feb 2026).

6. Limitations, controversies, and open directions

Several recurring issues determine whether an anisotropic diffusion model is physically credible or numerically reliable. One is identifiability. In the stress-coupled model, parametric identification of $D_i(t)$1 is explicitly listed as an open problem requiring experimental data such as optical mapping under stretch (Cherubini et al., 2017). In inverse reconstruction, the Levenberg–Marquardt procedure is computationally expensive because each quasi-Newton step requires solving $D_i(t)$2 sensitivity ODEs (Luchesi, 15 Sep 2025). In the matrix-valued schedule framework for diffusion models, the closed-form updates rely on a commuting-matrix assumption, and the gradient estimator requires higher-order directional derivatives implemented by multiple backward passes (Liu et al., 23 Feb 2026).

A second issue is whether anisotropy alone suffices to explain observed data. In TeV halos, the central observational tension is that random field orientations should often produce elongated halos, yet the detected halos appear nearly circular. The paper’s answer is a detectability bias: for $D_i(t)$3, one-year exposure, and $D_i(t)$4, Geminga-like halos exceed $D_i(t)$5 only if $D_i(t)$6 and $D_i(t)$7 kpc, with a typical critical viewing angle $D_i(t)$8–$D_i(t)$9 for plausible parameters and $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$00 yr (Yan et al., 2022). After $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$01 yr, LHAASO can push $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$02 to $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$03–$$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$04 and detect asymmetric halos out to $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$05–2 kpc for $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$06; conversely, a robust non-detection of elongated halos after a few years would strongly disfavor the anisotropic diffusion scenario (Yan et al., 2022). This is one of the clearest examples in which anisotropy generates a falsifiable morphological prediction rather than merely improving fit quality.

A third issue is regularization and admissibility under strong anisotropy. For non-convex surface energies, the doubly degenerate anisotropic Cahn–Hilliard model adds a Willmore-type term, leading to a sixth-order PDE (Salvalaglio et al., 2020). In numerical random-walk models, admissible probabilities require explicit constraints on $$\frac{\partial n}{\partial t} = \nabla\!\cdot\Bigl[D_{\parallel}\,\hat b\,\hat b + D_{\perp}\,\bigl(\mathbf I-\hat b\,\hat b\bigr)\Bigr]\nabla n + Q(\mathbf r,t,E_e),$$07, and on rectangular lattices these constraints imply a restriction on the diffusion tensor, whereas hexagonal lattices avoid that limitation (Filippini et al., 17 Oct 2025). In radiative transfer, previous failures of anisotropic diffusion theory are attributed not to the diffusion approximation itself but to inadequate boundary conditions or incorrect microscopic-to-macroscopic tensor relations; with the corrected derivation, the paper states that previous claims against anisotropic diffusion theory are falsified (Alerstam, 2013).

Taken together, these results indicate that anisotropic diffusion is best understood not as a single technique but as a modeling principle: diffusion is made direction-sensitive in a way that reflects field alignment, stress, structural coherence, graph geometry, subspace allocation, or anisotropic surface energy. The common gains are improved physical fidelity, sharper morphology, and greater control over propagation direction. The common costs are stronger identifiability requirements, more complex stability conditions, and a heavier dependence on geometry-aware numerics.